We consider a Celestial Mechanics model: the spin-orbit problem with a dissipative tidal torque, which is a singular perturbation of a conservative system. The goal of this paper is to show that it is possible to compute quasi-periodic attractors accurately and reliably for parameter values extremely close to the breakdown. Therefore, it is possible to obtain information on mathematical phenomena at breakdown. The method we use incorporates the same time numerical and rigorous improvements. Among them (i) the formalism is based on studying the time-one map of the spin-orbit problem (which reduces the dimensionality of the problem) and has mathematical advantages; (ii) very accurate integration of the ODE (high order Taylor methods implemented with extended precision) for the map at its jets; (iii) a very efficient KAM method for maps which computes the attractor and its tangent spaces ( quadratically convergent step with low storage requirements, and low operation count); (iv) the algorithms are backed by a rigorous a-posteriori KAM Theorem, which establishes that if the algorithm, produces a very approximate solution of functional equation with reasonable condition numbers. then there is a true solution nearby; and (v) the continuation algorithm is guaranteed to reach arbitrarily close to the border of existence if it is given enough computer resources. As a byproduct of the accuracy that we maintain till breakdown, we study several scale invariant observables of the tori used in the renormalization group of infinite dimensional spaces. In contrast with previously studied simple models, the behavior at breakdown of the spin-orbit problem does not satisfy standard scaling relations which implies that the spin-orbit problem is not described by a hyperbolic fixed point of a renormalization operator.

翻译：我们考虑一个天体力学模型：具有耗散潮汐力矩的自旋-轨道问题，该模型是保守系统的奇异摄动。本文旨在证明，对于极接近破裂的参数值，仍可精确可靠地计算准周期吸引子，从而获取破裂时刻的数学现象信息。所采用的方法融合了数值计算与严格理论的改进，具体包括：(i) 基于自旋-轨道问题的时间一映射（降低问题维度）的形式化体系，具有数学优势；(ii) 利用高精度泰勒方法及扩展精度对映射及其射流进行极高精度的ODE积分；(iii) 针对映射设计的高效KAM方法，可计算吸引子及其切空间（二次收敛步骤，低存储需求与低运算量）；(iv) 以严格的先验KAM定理为算法支撑，该定理指出若算法在合理条件数下生成函数方程的高度近似解，则其邻近必然存在真实解；(v) 连续算法在充足计算资源下可保证任意接近存在域边界。作为维持至破裂点精度的副产品，我们研究了无穷维空间重正化群中环面的若干标度不变可观测量。与先前研究的简单模型不同，自旋-轨道问题在破裂点的行为不满足标准标度关系，表明该问题并非由重正化算子的双曲不动点所描述。